Sampling Inspection Plan to Test Daily COVID-19 Cases Using Gamma Distribution under Indeterminacy Based on Multiple Dependent Scheme

The purpose of this paper is to develop a multiple dependent state (MDS) sampling plan based on time-truncated sampling schemes for the daily number of cases of the coronavirus disease COVID-19 using gamma distribution under indeterminacy. The proposed sampling scheme parameters include average sample number (ASN) and accept and reject sample numbers when the indeterminacy parameter is known. In addition to the parameters of the proposed sampling schemes, the resultant tables are provided for different known indeterminacy parametric values. The outcomes resulting from various sampling schemes show that the ASN decreases as indeterminacy values increase. This shows that the indeterminacy parameter plays a vital role for the ASN. A comparative study between the proposed sampling schemes and existing sampling schemes based on indeterminacy is also discussed. The projected sampling scheme is illustrated with the help of the daily number of cases of COVID-19 data. From the results and real example, we conclude that the proposed MDS sampling scheme under indeterminacy requires a smaller sample size compared to the single sampling plan (SSP) and the existing MDS sampling plan.


Introduction
Nowadays, most of the countries in the world are affected by the current COVID-19 pandemic. COVID-19 is the infectious disease caused by the most recently discovered coronavirus. The most common symptoms of COVID-19 are fever, tiredness, and a dry cough. In more severe cases, the infection can cause pneumonia, severe acute respiratory syndrome, and even death. The number of cases in the pandemic is unknown in most countries worldwide. Aside from that, when cases are spreading, the usual practice of any country is to test the people who show more symptoms; on the other hand, there are some people that do not have any symptoms or only experience a few symptoms [1]. Through these less symptomatic people, the coronavirus spreads more in society. To identify these types of people, more health workers are employing the methodology of randomly testing chosen persons to approximately calculate the actual number of cases in a specified area and, hence, the total state. In such situations, an acceptance sampling plan under indeterminacy is a suitable alternative to testing or assessing the number of cases in a particular locality. Health workers endure pressure to approximate the average daily number of cases of COVID-19 at present and for the next few days, few weeks, or months. For more details, see [2]. The researchers or health workers are interested in testing the null hypothesis that the average daily number of cases is equivalent to the specified average daily number of cases of COVID-19 and the alternative hypothesis that the average daily

Multiple Dependent State Sampling Plan under Indeterminacy
In this section, the development of the MDS sampling plan is discussed. The following are the essential conditions of pertinence to the proposed MDS sampling plan (see [16]): (i) The inspection policy consists of taking successive lots manufactured from a continuous manufacturing process. This means that the lots are submitted for inspection serially in the order they have been manufactured in the manufacturing process; (ii) The submitted lots for the examination have, for all intents and purposes, the same quality level. This means that the manufacturing process has a constant non-conforming fraction; (iii) The consumer has assurance in the reliability of the manufacturer's manufacturing process. It means that there is not any basis to consider that any specific lot quality is of inferior quality to the previous lots; (iv) The quality attribute under contemplation follows a gamma distribution.
The MDS plan is an extension of the SSP. In the MDS plan, the lot-declaring scheme is developed from a one-critical-point to a two-critical-point plan, namely, a lot-accepted critical point c 1 and a lot-rejected critical point c 2 , which allows the experiential quality level in between (c 2 , c 1 ) to judge the past, m-lot quality history. Based on this well-versed review, the MDS plan gains an advantage, economically, from governing smaller samples than the SSP.
The following, well-designed methodology for the MDS sampling design was given by [16] under neutrosophic statistics suggested by [44].
Step 1: Select a sample of size n from the batch. These specimens are employed for a life test for a predetermined time t N0 . Stipulate the average µ 0N , and indeterminacy quantity is I N [I L , I U ].
Step 2: The test H 0 : µ N = µ 0N can be accepted if the average daily number of cases for c 1 days is greater or equal to µ 0 (i.e., µ 0N ≤ c 1 ). If the average daily number of cases in c 2 days is less than µ 0 (i.e., µ 0 > c 2 ), then test H 0 : µ N = µ 0N can be rejected, and the test can be ended where c 1 ≤ c 2 .
Step 3: When c 1 < µ 0N ≤ c 2 , then accept the current lot provided that, in m preceding lots, the mean number of cases is less than or equal to c 1 before the test termination time t N0 .
The planned MDS sampling plan under indeterminacy is totally differentiated by four values, namely, n, c 1 , c 2 , and m, where n is the sample size, c 1 is the maximum number of allowable items that failed for unconditional acceptance, c 1 , c 2 is the maximum number of additional items that failed for conditional acceptance c 1 ≤ c 2 , and m is the number of successive lots (previous) needed to make a decision. The attributes' MDS sampling plan converges to m → ∞ and/or c 1 = c 2 = c (say), and MDS is an oversimplification of SSP. The operating characteristic (OC) function can reveal the concert of any sampling design.
Applying binomial chance law, the OC function for the MDS sampling design based on GD is expressed as follows [16]: where and where T is a random variable. Thus, the final expression for the OC function for MDS sampling design is: The chance of lot approval is obtained at failure probability p under binomial probability distribution.
Suppose that t N [t L , t U ] is the neutrosophic random variable that follows the gamma distribution. By following [44], let us assume that is a neutrosophic probability density function (npdf) with the determinate part f (t L ), indeterminate part f (t U )I N , and indeterminacy period I N [I L , I U ]. Note that the measure of indeterminacy I N [I L , I U ] presents uncertainty in the observations and parameters under an uncertain environment. Remember that t N [t L , t U ] considers a neutrosophic random variable (nrv) which abides by the npdf. The npdf is the oversimplification of pdf based on traditional figures. Thus, the planned neutrosophic form of ] becomes the pdf of traditional figures as soon as I L = 0. Using this information, the npdf and neutrosophic cumulative distribution function (ncdf) of the GD is determined as under: and where Γ(γ, t N /θ) is the lower incomplete gamma function, and (γ, θ). is the shape and scale parameters. The average lifetime of the neutrosophic GD is µ 0N = γθ + γθ I N . A product failure probability before the time t N0 is denoted as p N = F(t N ≤ t N0 ) and is conveyed below: Here, the neutrosophic termination time t N0 is express as product of constant a and neutrosophic mean life µ 0N , i.e., t N0 = aµ 0N . The scale parameter θ can be expressed in terms of the neutrosophic mean µ 0N .
Therefore, Equation (4) can be rewritten in terms of the neutrosophic mean µ 0N as follows: where µ N /µ 0N is the ratio of the exact average to the stipulated average. Suppose that α and β are the probabilities of type-I and type-II errors. The researcher should pay attention to the projected plan when examining H 0 : µ N = µ 0N in order to calculate the chance of accepting H 0 : µ N = µ 0N as soon as the true quantity becomes at least 1 − α at µ/µ 0 and the chance of accepting H 0 : µ N = µ 0N as soon as the false quantity becomes at most β at µ N /µ 0N = 1. The plan constants for examining H 0 : µ N = µ 0N are determined in such a way that the below two inequalities are fulfilled.
where p 1N and p 2N are defined by and Frequently, on-hand sampling schemes are intended to minimize the ASN. Commonly, the foremost intention of any sampling design is to decrease the ASN, which, in turn, minimizes both time and cost for inspection. Correspondingly, the projected MDS sampling design is intended to diminish the ASN for GD for the proposed situation. The non-linear programming method is adopted to get the optimal quantities, and it is expressed as follows: Minimize where p 1N and p 2N are the failure probabilities of the producer's and consumer's risks, respectively. These acceptance chances can be found by means of the following expressions: and The proposed plan consists of parameters c 1 , c 2 , m and ASN is obtained by solving the non-linear programming problem in Equation (11) for β = {0.25, 0.10, 0.05}, α = 0.10, and a = 0.5, 1.0, and known I N is placed in Tables 1-8. Tables 1 and 2 show the GD for γ = 2, Tables 3 and 4 show the GD for γ = 2.5, Tables 5 and 6 for γ = 3, Tables 5 and 6 for γ = 3, and Tables 7 and 8 for γ = 1. (exponential distribution). From the results in the tables, the following few points can be noticed:  Tables 7 and 8, it can be observed that GD shows a small ASN compared with exponential distribution; (f) Furthermore, it is depicted by the OC curves that the proposed MDS sampling plan under indeterminacy is more efficient than the existing single sampling plan.

Comparative Studies
This section deals with a comparative study of the proposed sampling scheme with the existing SSP. The efficiency of the developed sampling plan is calculated based on the ASN; a low-sample-size design is more economical to test the hypothesis about the mean. It is important to note that the proposed MDS sampling plan under indeterminacy is the generalization of the MDS sampling plan for traditional statistics if no uncertainty or indeterminacy happens when measuring the average. If I N = 0, the proposed MDS sampling plan under indeterminacy becomes the MDS sampling plan in hand. In Tables 1-8, the first column, i.e., at I N = 0, is the plan parameter of the traditional or existing MDS sampling plan. From the results, we conclude that the ASN is large in the traditional MDS sampling plan compared with the proposed MDS sampling plan. For example, when α = 0.10, β = 0.25, µ N /µ 0N = 1.4, γ = 2, and a = 0.5, from Table 1 it can be seen that ASN = 39 from the plan under classical statistics, and ASN = 34 for the projected sampling plan when I N = 0.05. Furthermore, when γ = 1, the GD becomes an exponential distribution; we constructed Tables 7 and 8 for exponential distribution for comparison purposes. Table 7 depicts that the GD shows a lower sample number compared with exponential distribution. For example, when α = 0.10, β = 0.25, µ N /µ 0N = 1.5, a = 0.5, and I N = 0.04, Table 7 shows that the ASN is 42, whereas the proposed plan values are ASN = 25 for γ = 2, ASN = 24 for γ = 2.5, and ASN = 20 for γ = 3. From this study, it is concluded that the projected plan under indeterminacy is more efficient than the existing sampling plan under traditional statistics with respect to sample size. The operating characteristic (OC) curve of the plan of the GD when α = 0.10, β = 0.10, γ = 3.0, and a = 0.50 is depicted in Figures 1 and 2. Therefore, the application of the proposed plan for testing the null hypothesis H 0 : µ N = µ 0N demands a smaller sample compared to the on-hand plan. The OC curve in Figure 1 also shows the same performance. Researchers can apply the proposed plan under uncertainty to save time and money. when = 0.10, = 0.25, μ μ ⁄ = 1.4, = 2, and = 0.5, from Table 1 it can be seen that = 39 from the plan under classical statistics, and = 34 for the projected sampling plan when = 0.05. Furthermore, when = 1, the GD becomes an exponential distribution; we constructed Tables 7 and 8 for exponential distribution for comparison purposes. Table 7 depicts that the GD shows a lower sample number compared with exponential distribution. For example, when = 0.10, = 0.25, μ μ ⁄ = 1.5, = 0.5, and = 0.04, Table 7 shows that the ASN is 42, whereas the proposed plan values are ASN = 25 for = 2, ASN = 24 for = 2.5, and ASN = 20 for = 3. From this study, it is concluded that the projected plan under indeterminacy is more efficient than the existing sampling plan under traditional statistics with respect to sample size. The operating characteristic (OC) curve of the plan of the GD when = 0.10, = 0.10, = 3.0, and = 0.50 is depicted in Figures 1 and 2. Therefore, the application of the proposed plan for testing the null hypothesis H : μ = μ demands a smaller sample compared to the on-hand plan. The OC curve in Figure 1 also shows the same performance. Researchers can apply the proposed plan under uncertainty to save time and money.
It is established that the daily number of cases data can be drawn from the GD with shape parameterγ = 3.1946 and scale parameterθ = 7.0809, and the maximum distance between the real-time data and the fitted of GD can be found from the Kolmogorov-Smirnov test statistic 0.0803 and also the p-value 0.8262. The demonstration of the goodness of fit for the given model, the empirical and theoretical pdfs, cdf, P-P plot and Q-Q plots for the GD for the daily number of cases data are shown in Figure 3. Therefore, GD is well fitted for the daily number of cases of COVID-19 data. The plan parameters for this shape parameter are shown in Tables 9 and 10. For the proposed plan, the shape parameter iŝ γ N = (1 + 0.04) × 3.1946 ≈ 3.3224 when I U = 0.04. Suppose that medical administrators are concerned with testing H 0 : µ N = 23.5255 with the aid of the proposed sampling plan when I U = 0.04, α = 0.10, µ N /µ 0N = 1.4, a = 0.5, and β = 0.10. From Table 9, it can be noted that n = 55, c 1 = 6, c 2 = 14, m = 1, and ASN = 55. The developed MDS sampling plan works by accepting the null hypothesis H 0 : µ N = 23.5255 if the average daily number of cases in 6 days is more than equal to 23.5255 daily number of cases. A sample of 55 daily due to COVID-19 can be selected at random for a crowd of people, and the null hypothesis H 0 : µ N = 23.5255. If the average daily number of cases before 23.5255 is less than or equal to 6 days, then the crowd of people can be accepted, and the crowd of people can be rejected if it is greater than 14 days. If the prevalence of the number of cases of COVID-19 is between 6 and 14 days, a property of the present crowd of people can be deferred until the preceding crowd of people has be tested. From the data, it can be noted that an average daily number of cases of COVID-19 was greater than equal to 23.5255 encounter in more than 36 days; therefore, the claim about the average daily number of cases H 0 : µ N = 23.5255 could be rejected. Hence, medical administrators could suggest to the government that the average daily number of cases of COVID-19 was at an unendurable stage. Therefore, the proposed sampling plan is useful in medical applications, specifically, in taking decisions regarding the daily number of cases of COVID-19 and the average daily number of COVID-19 patients, and this is very important for any government making policy decisions.

Conclusions
A broad analysis of the daily number of cases of COVID-19 for gamma distribution based on indeterminacy situation for a time-truncated MDS sampling design was formulated. The sampling plan's quantities were determined at pre-assigned values of indeterminacy parameters. Comprehensive tables were given for ready reference to the researchers for the known indeterminacy constant values. The formulated MDS sampling design based on indeterminacy was compared with the already available sampling schemes based on classical statistics. The results showed that the formulated MDS sampling plan under indeterminacy was more reasonable than the already available SSP under indeterminacy and traditional MDS sampling plans. In addition, the developed MDS under indeterminacy was greatly cheaper to run than the SSP. It is important to note that the indeterminacy parameter showed a prime role in decreasing ASN values, which means that, if the indeterminacy value increased then the ASN values were in decreasing trend. Therefore, the formulated MDS sampling plan under indeterminacy is more useful to scientists, in particular to medical practitioners and those who are studying or testing sensitive issues which require skilled researchers and need more money. Thus, the formulated MDS sampling plan under indeterminacy is approved to be applicable for testing the average daily number of cases of COVID-19. The exemplification based on the daily number of cases of COVID-19 data for the formulated MDS sampling plan under indeterminacy showed confirmation. The formulated MDS sampling plan under indeterminacy can be used by other researchers working in various fields. Considering a control chart methodology based on multiple dependent state sampling plans will be the topic of a further research study to monitor the mean.